Problem: Simplify the following expression and state the condition under which the simplification is valid: $p = \dfrac{q^2 + 11q + 30}{q^2 + 4q - 5}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{q^2 + 11q + 30}{q^2 + 4q - 5} = \dfrac{(q + 6)(q + 5)}{(q - 1)(q + 5)} $ Notice that the term $(q + 5)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(q + 5)$ gives: $p = \dfrac{q + 6}{q - 1}$ Since we divided by $(q + 5)$, $q \neq -5$. $p = \dfrac{q + 6}{q - 1}; \space q \neq -5$